Eigenvectors of Some Large Sample Covariance Matrix Ensembles

Olivier Ledoit and Sandrine Péché

Abstract

We consider sample covariance matrices S_N = (1/p) Σ_N^1/2X_NX_N^*Σ_N^1/2 where X_N is a N × p real or complex matrix with i.i.d. entries with finite 12^th moment and Σ_N is a N × N positive definite matrix. In addition we assume that the spectral measure of Σ_N almost surely converges to some limiting probability distribution as N goes to infinity and p/N converges to γ > 0.We quantify the relationship between sample and population eigenvectors by studying the asymptotics of functionals of the type (1/N)Tr(g(Σ_N)(S_N-zI)^-1), where I is the identity matrix, g is a bounded function and z is a complex number. This is then used to compute the asymptotically optimal bias correction for sample eigenvalues, paving the way for a new generation of improved estimators of the covariance matrix and its inverse.